# Homework Help: Quantum scattering

1. Jun 12, 2009

### cleggy

1. The problem statement, all variables and given/known data

A beam of particles, each of mass m and energy E0,is incident on a square potential energy well of width L and depth V0,where 0 <V0 < h(bar)^2$$\pi$$^2/2mL^2 . Outside the region of the well, the potential energy is equal to zero. Suppose that E0 is the lowest energy at which a transmission resonance occurs, with 100% of the beam being transmitted and none reflected.

(a)
A second beam of particles, each of mass m and energy E1 is incident
on a square potential energy barrier of width L and height V0. Outside the
region of the barrier, the potential energy is equal to zero. What is the
lowest value of E1 at which a transmission resonance occurs in this
situation, with 100% of the beam being transmitted and none reflected?
(b)
A third beam of particles, each of mass m and energy V0/2is
incident on a square potential energy barrier of width L and height V0.
Outside the region of the barrier, the potential energy is equal to zero.
Estimate the fraction of particles that tunnels through the barrier if
V0 =1.0h(bar)^2/mL^2 .
(c)
Suppose that the barrier in part (b) extends from x =0 to x = L and the incident beam travels in the positive x-direction. For x< 0 the energy eigenfunction describing the beam is Aexp(ikx) + Bexp(−ikx), while for x>L it is Fexp(ikx),where k is the wave number and A, B and F are constants. For the conditions described in part (b), what is the value of the ratio |F|/|B|?

2. Relevant equations

3. The attempt at a solution

Can someone give me help with starting this please?

Is the lowest energy E0 = V0/2 ?

if so how is this derived?

Last edited: Jun 12, 2009
2. Jun 13, 2009

### drizzle

first you need to sketch the problem, so you can understand it well [I've done it for you], apparently you have three zones as shown bellow

in a general matter, you have to solve the Schrödinger equation in the x direction, you should have 3 equations for this system [one for each zone], you also need to consider the primary conditions to solve it.

hint: with the beam being 100% transmitted, what do you think the energy of the particles in this system should be, E>Vo or E<Vo?

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